Minimal Systems on Cantor Set Maryam Hosseini Thematic Program in - PowerPoint PPT Presentation

Minimal Systems on Cantor Set Maryam Hosseini Thematic Program in Dynamical Systems IPM, Tehran February 2017

Outline 1 Introduction 2 Examples: Odometers Substitutions Toeplitz 3 Kakutani-Rokhlin towers for minimal systems Bratteli diagrams and Vershik Systems with Examples Some Dynamical Properties of Vershik Homeomorphisms

Introduction Minimal systems: Natural generalizations of periodic orbits and topological analogous of ergodic systems, defined by [G. D. Birkhof, 1912]. Extension to Cantor set: Theorem. (P. Alexandroff, 1927) Every compact metric space is a continuous image of the Cantor set. Let ( X, T ) be a minimal system on a compact metric space. ∃ F : C → X, C is the Cantor set which is continuous and onto. Set K := { ( x n ) n ∈ Z ; x n ∈ C, F ( x n +1 ) = T ( F ( x n )) } .

... K = { ( x n ) n ∈ Z ; x n ∈ C, F ( x n +1 ) = T ( F ( x n )) } ⊆ C Z and is σ -invariant. Let Z be a minimal subset of ( K, σ ) . Then ψ : ( Z, σ ) → ( X, T ) ψ (( z n ) n ∈ Z )) = z 0 makes the factoring. Remark. Note that Z is a closed subset of the C Z and so is a Cantor set.

1. Odometers (adding Machines) Let J = ( j 1 , j 2 , · · · ) be a sequence of natural numbers and X = { ( x n ) n ∈ N 0 : 0 ≤ x i ≤ j i − 1 } . The adding machine is defined by the map T : X → X with T ( x 0 , x 1 , · · · ) = ( x 0 , x 1 , · · · ) + (1 , 0 , 0 , · · · ) . The addition is component-wise with carrying to the right. This system is minimal and distal, means that d ( T n x, T n y ) > δ, ∀ n ≥ 0 . ∀ x, y ∈ X, ∃ δ > 0; In fact, δ x,y = d ( x, y ) . In fact, it is equicontinuous , means that { T n } n is an equicontinuous family.

... Theorem. (See [P. kurka 2003]) Every minimal equicontinuous system on Cantor set is conjugate to an odometer. proof. It suffices to consider the equivalent metric n d ( T n x, T n y ) . d ( x, y ) = sup Corollary. The maximal equicontinuous factor of a minimal distal system on Cantor set is conjugate to an odometer.

... Let n i := j i j i − 1 · · · j 1 . It’s pretty clear that T n i → id , or T n i x → x. ∀ x ∈ X, This property is called rigidity along the sequence { n i } i . Proposition. (E. Glasner, D. Maon, 1975) Any (infinite) minimal rigid system on Cantor set is conjugate to an odometer. Proof. Exercise (Hint: show that it is equicontinuous). Odometers are also called rotations or Kronecker system on Cantor set as they are isometries.

Odometers from algebraic point of view Let ( p i ) i ≥ 1 be a sequence of natural numbers that ∀ i ≥ 1 , p i ≥ 2 , p i | p i +1 . Consider the following inverse limit system: φ 1 φ 2 ( Z p 1 , ı 1 ) ← − ( Z p 2 , ı 2 ) ← − · · · ← − ( Z, ı ) where ı i ( z ) = z + 1 (mod p i ) and Z = { ( z n ) n ∈ N ; z n ∈ Z p n , φ i ( z i ) = z i (mod p i − 1 ) } and ı ( z 1 , z 2 , · · · ) = ( z 1 , z 2 , z 3 , · · · ) + (1 , 1 , 1 , · · · ) . Exercise. Show that ( Z, ı ) is conjugate to the odometer based on the sequence ( p i /p i − 1 ) i .

2. Substitutions Let A be a set of alphabets, like A = { 1 , 2 , . . . , k } and A + be the set of words with letters in A . A substitution on A is a map τ : A → A + that ∀ a ∈ A, | τ n ( a ) | → ∞ . By concatenation, one can extend such a map to A + : ∀ w = w 1 w 2 . . . w k ∈ A + , τ ( w ) = τ ( w 1 ) τ ( w 2 ) . . . τ ( w k ) . So τ n : A → A + is also a substitution, n times � �� � τ n ( a ) = τ n − 1 ( τ ( a )) = · · · = ∀ a ∈ A, τ ( τ ( · · · ( τ ( a ) · · · ) . A substitution is primitive if ∀ a, b ∈ A, ∃ p > 0; a appears in τ p ( b ) . Fixed points of a substitution: { x ∈ X τ : τ ( x ) = x } .

... Example i) Let A = { 0 , 1 } and τ (0) = 001 , τ (1) = 01 . Then τ τ τ τ �− → 001 �− → 00100101 �− → 001001010010010100101 �− → · · · ; 0 τ τ τ τ 1 �− → 01 �− → 00101 �− → 0010010100101 �− → · · · . Example ii)(Thue-Morse) Let A = { 0 , 1 } and τ (0) = 01 , τ (1) = 10 . Then τ τ τ τ 0 �− → 01 �− → 0110 �− → 01101001 �− → · · · , τ τ τ τ 1 �− → 10 �− → 1001 �− → 10010110 �− → · · · . Example iii) Let A = { 0 , 1 , 2 } . Then 0 �− → 01 , 1 �− → 2 , 2 �− → 012 Example iv) Let A = { 0 , 1 } . Then 0 �− → 010 , 1 �− → 111 .

... If there exists at least one letter a ∈ A so that τ ( a ) begins with a , then we have at least one fixed point. Definition. ∀ x ∈ A Z , L ( x ) = { u ∈ A + ; ∃ p > 0 , u ≺ τ p ( x ) } . It is easy to see that for a primitive τ , x, y ∈ A, τ ( x ) = x, τ ( y ) = y ⇒ L ( x ) = L ( y ) . Definition. A primitive substitution is proper if it has a unique fixed point. Remark. If ∃ r, ℓ ∈ A such that ∀ a ∈ A, τ ( a ) starts with r and ends with ℓ and rℓ is admissible then τ is proper.

Substitution dynamical systems Definition. Let X τ be a subset of A Z associated to the language of the fixed points of a primitive τ , i.e. X τ = { x ∈ A Z : ∀ i < j, x i x i +1 · · · x j ∈ L ( a ); a = τ ( a ) } . X τ together with the restriction of the shift map σ is called a Substitution dynamical system, ( X τ , σ ) . In other words, a subshift ( X, σ ) with the alphabet A, is a substitution if ∃ a primitive τ : A → A + , w = τ ( w ); X τ = { σ n ( w ) } n , Proposition. (F. Durand, B. Host, C. Skau, 1999) Every substitution dynamical system is conjugate to the closure orbit of the fixed point of a proper substitution.

Systems associated to sequences Let u = ( u n ) n be a sequence in a shift space and set X = { σ n ( u ) } n . Proposition. (See [M. Queffelece ’87] ) ( X, σ ) is minimal iff u is uniformly recurrent. Recall that uniform recurrence means that for any words w the set of gaps between any two consecutive occurrences of w is bounded. Corollary. Every substitution dynamical system, ( X, σ ) is minimal.

... Let u = ( u n ) n be a sequence in a shift space and ℓ B ( C ) be the number of occurrence of B in C , where B and C are two admissible words. We say that u has uniform word frequencies if ℓ B ( u k . . . u k + n ) ∀ B : lim n + 1 n →∞ exists uniformly in k (independent from k ). Proposition. (See [M. Queffelec ’87]) ( X, σ ) associated to the sequence u is uniquely ergodic iff u has uniform word frequencies. Hint. Use point-wise ergodic theorem.

The invariant measure Corollary. Every substitution dynamical system, ( X, σ ) is uniquely ergodic. In fact, for the substitution system ( X τ , σ ) with alphabet A, for every a ∈ A the map µ defined by 1 � µ := lim δ T n u | τ j ( a ) | j →∞ n< | τ j ( a ) | is an invariant Borel measure for the system which is unique.

Linear complexity Proposition. (See [M. Queffelec ’87]) Every substitution dynamical system has zero entropy. Proof. Consider the incidence matrix of the substitution . Using Perron-Frobenius Theorem, for the fixed point u , there exists r > 0 such that 1 p u ( n ) ≤ rn ⇒ lim n log( p u ( n )) = 0 . n →∞ Example i) Sturmian systems are substitutions or generated by finitely many substitutions. These are almost one to one extensions of irrational rotations on the unit circle with p u ( n ) = n + 1 .

(weakly) mixing substitution Example ii) Chacon’s minimal weakly mixing and non-mixing substitution system ( X, σ ) , where X is the orbit closure of the first fixed point of the following substitution: 0 �− → 0010 , 1 �− → 1 , which is non-primitive. But there exists a primitive substitution with 3 symbols that makes a conjugate system. Example iii) Dekking’s and Kean’s topologically mixing substitution system coming from: 0 �− → 001 , 1 �− → 11100 . Remark. (Dekking, Kean, 1978) A substitution can never be strongly mixing with respect to its unique invariant measure.

3. Toeplitz sequence, See [P. Kurka 2003] A point x in dynamical system ( X, T ) is quasi-periodic if ∀ U open set ; x ∈ U, ∃ p > 0; T np ( x ) ∈ U, ∀ n ≥ 1 . Recall that in odometers all points are quasi-periodic. Definition. A point x ∈ A N is Toeplitz if there exists an increasing sequence ( p i ) i ≥ 0 , p i ∈ N such that p i | p i +1 , for every n ≥ 0 there exists some i so that n ∈ per p i ( x ) , where per p i ( x ) = { k ∈ N : ∀ n x k + pn = x k } . So any Toeplitz sequence is quasi-periodic (w.r.t. shift map).

... The p -skeleton of x , S p ( x ) , is defined by � x i if i ∈ per p ( x ) S p ( x ) = ∈ per p ( x ) . * if i / So to construct the toeplitz sequence we need the ( p i ) i ≥ 0 , r i := min { k : k ∈ per p i ( x ) } . to find S p i ( x ) . Example. Let A = { 0 , 1 } and construct the toeplitz sequence with the periodic structure ( p n ) n = (2 n ) n ≥ 1 and r 2 = 0 , r 4 = 1 r 8 = 3 , r 16 = 7 , · · · . Then S 1 ( x ) = ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ · · · S 2 ( x ) = 1 ∗ 1 ∗ 1 ∗ 1 ∗ · · · ∗ ∗ · · · S 4 ( x ) = 1 0 1 1 0 1 S 8 ( x ) = 1 0 1 1 1 0 1 ∗ · · · · · · S 16 ( x ) = 1 0 1 1 1 0 1 0